The Stacks project

Proof. By Algebra, Lemma 10.78.5 we see that $M_\mathfrak p$ is finite free for all prime ideals $\mathfrak p \subset R$. By Algebra, Lemma 10.78.2 it suffices to show that the function $\rho _ M : \mathfrak p \mapsto \dim _{\kappa (\mathfrak p)} M \otimes _ R \kappa (\mathfrak p)$ is locally constant on $\mathop{\mathrm{Spec}}(R)$. Because $M/IM$ is finite projective, this is true on $V(I) \subset \mathop{\mathrm{Spec}}(R)$. Since every closed point of $\mathop{\mathrm{Spec}}(R)$ is in $V(I)$ and since $\rho _ M(\mathfrak p) = \rho _ M(\mathfrak q)$ whenever $\mathfrak p \subset \mathfrak q \subset R$ are prime ideals, we conclude by an elementary argument on topological spaces which we omit. $\square$